# Towards parametrizing word equations

• Volume: 35, Issue: 4, page 331-350
• ISSN: 0988-3754

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## Abstract

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Classically, in order to resolve an equation u ≈ v over a free monoid X*, we reduce it by a suitable family $ℱ$ of substitutions to a family of equations uf ≈ vf, $f\in ℱ$, each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get $ℱ$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to u ≈ v. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables.

## How to cite

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Abdulrab, H., Goralčík, P., and Makanin, G. S.. "Towards parametrizing word equations." RAIRO - Theoretical Informatics and Applications 35.4 (2010): 331-350. <http://eudml.org/doc/222084>.

@article{Abdulrab2010,
abstract = { Classically, in order to resolve an equation u ≈ v over a free monoid X*, we reduce it by a suitable family $\cal F$ of substitutions to a family of equations uf ≈ vf, $f\in\cal F$, each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get $\cal F$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to u ≈ v. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables. },
author = {Abdulrab, H., Goralčík, P., Makanin, G. S.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Equation; free monoid; parametrization; universal family.; universal family},
language = {eng},
month = {3},
number = {4},
pages = {331-350},
publisher = {EDP Sciences},
title = {Towards parametrizing word equations},
url = {http://eudml.org/doc/222084},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Abdulrab, H.
AU - Goralčík, P.
AU - Makanin, G. S.
TI - Towards parametrizing word equations
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 4
SP - 331
EP - 350
AB - Classically, in order to resolve an equation u ≈ v over a free monoid X*, we reduce it by a suitable family $\cal F$ of substitutions to a family of equations uf ≈ vf, $f\in\cal F$, each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get $\cal F$ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to u ≈ v. We carry out such a parametrization in the case the prime equations in the graph involve at most three variables.
LA - eng
KW - Equation; free monoid; parametrization; universal family.; universal family
UR - http://eudml.org/doc/222084
ER -

## References

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1. J. Jaffar, Minimal and Complete Word Unification. J. ACM37 (1990) 47-85.
2. Yu.I. Hmelevskii, Equations in free semigroups. Trudy Mat. Inst. Steklova 107 (1971); English Translation in Proc. Steklov Inst. Math. 107 (1971) 1976.
3. A. Lentin, Équations dans les monoïdes libres. Gautier-Villars, Paris (1972).
4. M. Lothaire, Combinatorics on Words. Addison-Wesley (1983).
5. G.S. Makanin, The problem of solvability of equations in a free semigroup. Mat. Sbornik 103 (1977) 147-236 (in Russian); English Translation in Math. USSR Sbornik 32 (1977) 128-198.
6. G.S. Makanin, On general solution of equations in free semigroups, in Proc. of IWWERT'91, edited by H. Abdulrab and J.P. Pécuchet. Springer, Lecture Notes in Comput. Sci. 677, 1-5.
7. G. Plotkin, Building-in Equational Theories. Machine intelligence7 (1972) 73-90.

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